1. Field of the Invention
The present invention relates to a computer-assisted analysis method for predicting the growth rate distribution of a plated film in electroplating to obtain a uniform plating thickness distribution. More particularly, the invention relates to a method preferred for analysis of the plating rate distribution of a metal intended for wiring on a semiconductor wafer.
2. Description of the Related Art
In a system in which an anode and a cathode constitute a cell via an electrolyte and form a potential field in the electrolyte, a potential distribution and a current density distribution are important for such a problem as a plating or corrosion problem. To predict these distributions in the system, computer-assisted numerical analysis by the boundary element method, the finite element method, or the finite difference method has been attempted. This analysis is conducted based on the facts that the potential in the electrolyte is dominated by Laplace""s equation; that the potential and current density on the anode surface and the cathode surface are ruled by an electrochemical characteristic, called a polarization curve (nonlinear functions found experimentally for showing the relationship between potential and current density), determined by a reaction caused when the anode and the cathode are disposed in the electrolyte; and that the current density is expressed as the product of a potential gradient and the electrical conductivity of the electrolyte.
In electroplating, the plating rate of a metal deposited on the cathode can be calculated from the analyzed current density of the cathode by Farady""s law. Thus, the above-mentioned numerical analysis enables the plating rate distribution to be predicted beforehand according to the conditions, such as the structure of a plating bath, the type of a plating solution, and the types of materials for the anode and the cathode. This makes it possible to design the plating bath rationally.
In recent years, it has been attempted to utilize electroplated copper for wiring in a semiconductor integrated circuit. In this case, as shown in FIG. 1A, fine grooves 2 are formed by etching in a surface of an interlayer insulator film 1 of SiO2 or the like on a semiconductor wafer W. Copper, a material for wiring, is buried in the grooves 2 by electroplating. To prevent mutual diffusion between the copper and the SiO2 film, a barrier layer 3 of TaN or the like is formed beforehand on the surface of the SiO2 film by a method such as sputtering. Since SiO2 and TaN are insulators or high resistance materials, a thin film (called a seed layer) 4 of copper, which acts as a conductor and an electrode for electroplating, is formed on the TaN by a method such as sputtering.
The seed layer 4 of copper formed beforehand is as thin as about several tens of nanometers in thickness. While a current is flowing through this thin copper seed layer, a potential gradient occurs in this seed layer because of its resistance. If plating is carried out with a layout as shown in FIG. 1A, a nonuniform thickness of plating, i.e., thick on the outer periphery and thin on the inner periphery, arises as shown by a solid line 5 in the drawing, since a current flows more easily nearer to the outer peripheral region. As shown in FIG. 1B, moreover, when a metal such as copper is buried in fine holes or fine grooves by plating, a potential gradient appears in the copper seed layer because of the resistance of the seed layer. As a result, the plating rate increases near the entrance of the hole or groove, and defects, such as portions void of copper, occur in the hole or groove. An additive for suppressing the reaction is used to bring down the preferential growth rate of a plating in the vicinity of the groove, thereby preventing the occurrence of internal defects.
Many conventional methods of plating analysis are based on the concept that a potential gradient occurs only in an electrolyte, and the resistances of an anode and a cathode are so low as to be negligible. In analyzing the current density distribution and the voltage distribution of electroplating on a semiconductor wafer, however, the resistance on the electrode side cannot be neglected, and needs to be considered.
An example of a plating analysis method taking the electrode-side resistance into consideration has been attempted by the finite element method. According to this method, the interior of a plating solution region is divided into elements. Resistance conditions for the plating solution are put into these elements, and the electrode with resistance is divided into elements as deposition elements. Resistance conditions for the electrode are put into these elements. Furthermore, an element called an overvoltage element is newly created at a position, on the surface of the electrode (mainly cathode), in contact with the plating solution. In this element, the conditions for polarization resistance of the electrode are placed. The entire element is regarded as a single region, and analyzed by the finite element method. The deposition elements correspond to a plated film. The thickness of the plated film at the start of plating is zero. Then, the film thickness determined by the current density calculated at elapsed time points is accumulated, and the values found are handled as the thickness.
A suitable structure of the plating bath and a suitable arrangement of electrodes are devised by numerical calculation or based on a rule of thumb. To make the plating rate uniform, placement of a shield plate in the plating solution for avoiding concentration of a current in the outer peripheral portion, for example, has been proposed and attempted. However, a sufficient effect has not been obtained. Nor has any rational method concerning a design of the shield plate been established up to now.
It is generally pointed out that the boundary element method requiring no element division of the interior is advantageous in analyzing problems (such as plating, corrosion and corrosion prevention problems) for which a potential distribution and a current density distribution on the surface of a material are important. The boundary element method is applied to the analysis of a plating problem requiring no consideration for the resistance of an electrode, and its effectiveness has already been confirmed. However, it has not been known that the boundary element method can be applied for a plating problem requiring consideration for the resistance of an electrode.
As described above, the finite element method has been applied to a plating problem requiring consideration for the resistance of an electrode. However, the finite element method requires the division of the interior into elements, thus involving a vast number of elements. Consequently, this method takes a long time for element division and analysis.
The present invention has been accomplished under these circumstances. An object of the invention is to provide a plating analysis method which can obtain a current density distribution and a potential distribution efficiently for a plating problem requiring consideration for the resistance of an electrode. Another object of the invention is to provide a plating analysis method for optimizing the structure of a plating bath designed to uniformize a current, which tends to be concentrated near an outer peripheral portion of a cathode, thereby making the plating rate uniform.
A first aspect of the present invention is a plating analysis method for electroplating in a system. The method comprises: giving a three-dimensional Laplace""s equation, as a dominant equation, to a region containing a plating solution between an anode and a cathode; discretizing the Laplace""s equation by a boundary element method; giving a two-dimensional or three-dimensional Poisson""s equation dealing with a flat surface or a curved surface, as a dominant equation, to a region within the anode and/or the cathode; discretizing the Poisson""s equation by the boundary element method or a finite element method; and formulating a simultaneous equation of the discretized equations to calculate a current density distribution and a potential distribution in the system.
According to this aspect, the Poisson""s equation is given to the region within the anode and/or the cathode in consideration of the resistance of the anode and/or the cathode. This ensures consistency with the region within the plating solution to be dominated by the three-dimensional Laplace""s equation. Thus, while the influence of the resistance of the anode and/or cathode is considered, the element division of the region within the plating solution is not necessary, so that the time required for element division and analysis can be markedly shortened. This aspect, therefore, enables accurate and efficient simulation of the current density distribution and the potential distribution within the plating bath that takes the influence of the resistance of the anode and/or the cathode into consideration.
The plating analysis method may further comprise giving the electrical conductivity or resistance of the anode and/or the cathode, as a function of time, to the region within the anode and/or the cathode. Thus, even if the resistance value distribution of the cathode, a semiconductor wafer as an object to be plated, changes because of deposition of a plated film on the cathode with the passage of time, it becomes possible to simulate the state of the change in the distribution.
The plating analysis method may further comprise: dividing the anode into two or more divisional anodes; and calculating such optimum values of current flowing through the divisional anodes as to uniformize a current density distribution on the surface of the cathode, thereby uniformizing the plating rate. This makes it possible to simulate the structure of the plating bath, the shape of the divisional anode, and the method for current supply that will apply a uniformly thick plated film onto the entire surface of a semiconductor wafer.
The plating analysis method may further comprise: calculating and giving the optimum values of current flowing through the divisional anodes at time intervals, thereby uniformizing the plating rate. Thus, simulation can be performed so that even when a thick plated film is applied over time, a uniform current density distribution is obtained on the entire surface of the wafer to obtain a uniform plated film thickness.
A second aspect of the invention is a plating apparatus produced with the use of any one of the plating analysis methods described above.
In the plating apparatus, the position, shape, and size of the anode and/or the position, shape and size of a shield plate may have been adjusted so that the current density distribution on the cathode surface will be uniformized by use of any one of the above plating analysis methods.
A third aspect of the invention is a plating method comprising: applying a metal plating by use of any one of the plating analysis methods described above, the metal plating being intended for formation of wiring on a wafer for production of a semiconductor device.
A fourth aspect of the invention is a method for producing a wafer for a semiconductor device, comprising: applying plating to the wafer by the plating method described above; and polishing the surface of the wafer by chemical and mechanical polishing (CMP) to produce the wafer of a desired wiring structure.
A fifth aspect of the invention is a method for analysis of corrosion and corrosion prevention in a system. The method comprises: giving a three-dimensional Laplace""s equation, as a dominant equation, to a region containing an electrolyte; discretizing the Laplace""s equation by a boundary element method; giving a two-dimensional or three-dimensional Poisson""s equation dealing with a flat surface or a curved surface, as a dominant equation, to a region within the anode and/or the cathode; discretizing the Poisson""s equation by the boundary element method or a finite element method; and formulating a simultaneous equation of the discretized equations to calculate a current density distribution and a potential distribution in the system.
This aspect enables the present invention to be used for analysis of corrosion and corrosion prevention.
To sum up the effects of the invention, the finite element method has been the only feasible method for numerical analysis of the plating rate distribution in electroplating of a system in which the resistance of an anode and/or a cathode is not negligible. However, when dividing the regions of the plating bath into elements, even the interior region needs to be divided, thus taking a vast amount of time for element division and analysis.
The methods of the present invention employing the boundary element method do not require element division within a plating solution, and thus can markedly shorten the time for element division and analysis. Moreover, when the shape of the plating bath is axially symmetrical and can be modeled, the region accounted for by the solution can be divided into axially symmetrical elements. Thus, more efficient analysis can be performed.
In connection with electroplating in a system in which the resistance of a cathode is not negligible, there has been a demand for a method, which can correct non uniformity of the plating rate due to the presence of resistance of the cathode. To satisfy this demand, the invention provides methods, which comprise dividing an anode suitably, and calculating optimal values of current to be flowed through the divisional anodes. These methods can uniformize the current, which tends to be concentrated in the peripheral portion of the cathode, by a short time of analysis.